The speeds of four cars are 2u, 3u, 4u and xu and the time taken by them to cover the same distance is xt, 4t, 3t and 2t respectively, where x, u, t

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CDS - 2024

The speeds of four cars are \(2u, 3u, 4u\) and \(xu\) and the time taken by them to cover the same distance is \(xt, 4t, 3t\) and \(2t\) respectively, where \(x, u, t\) are real numbers. What is the value of \(x\)?

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CDS 2 2024 Maths Question Paper (01-Sep-2024)

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The correct answer is

Solving for Car Speed Variable 'x' Using Distance Formula

This solution explains how to find the unknown variable 'x' related to the speeds and times of four cars covering the same distance. We utilize the fundamental relationship between speed, distance, and time.

Understanding the Relationship: Distance, Speed, and Time

The core principle connecting these variables is:

\( \text{Distance} = \text{Speed} \times \text{Time} \)

The problem specifies that all four cars travel the same distance. This key piece of information allows us to set up equations by equating the distance calculated for each car.

Setting up the Equations for Car Speeds and Times

Here's a breakdown of the information provided for each car:

Car 1: Speed = \(2u\), Time = \(xt\)
Car 2: Speed = \(3u\), Time = \(4t\)
Car 3: Speed = \(4u\), Time = \(3t\)
Car 4: Speed = \(xu\), Time = \(2t\)

Let the constant distance covered by each car be denoted by \(D\). We can express \(D\) for each car using the formula:

Car 1: \( D = (2u) \times (xt) = 2uxt \)
Car 2: \( D = (3u) \times (4t) = 12ut \)
Car 3: \( D = (4u) \times (3t) = 12ut \)
Car 4: \( D = (xu) \times (2t) = 2xut \)

Calculating the Value of x

Since the distance \(D\) is constant for all cars, we can equate these expressions. A straightforward approach is to equate the distance calculated for Car 1 with the distance calculated for Car 2:

\( 2uxt = 12ut \)

To solve for \(x\), we simplify this equation. Assuming \(u\) and \(t\) are non-zero real numbers (as they represent speed and time), we can divide both sides of the equation by \(ut\):

\( 2x = 12 \)

Now, we isolate \(x\) by dividing both sides by 2:

\( x = \frac{12}{2} \)

\( x = 6 \)

Verifying the Solution

To ensure consistency, let's check if \(x=6\) works with the data for Car 4. The distance for Car 4 is \(D = (xu) \times (2t)\). Substituting \(x=6\):

\( D = (6u) \times (2t) = 12ut \)

This calculated distance (\(12ut\)) perfectly matches the distance derived from Car 2 and Car 3 (\(12ut\)). This confirms that our calculated value \(x=6\) satisfies all conditions of the problem.

Conclusion

The value of \(x\) that satisfies the given conditions regarding the speeds and times of the four cars covering the same distance is 6.

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Solving for Car Speed Variable 'x' Using Distance Formula

Understanding the Relationship: Distance, Speed, and Time

Setting up the Equations for Car Speeds and Times

Calculating the Value of x

Verifying the Solution

Conclusion

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